Let $X$ be a complex smooth quasi-projective surface acted upon by a finite group $G$ such that the quotient $X/G$ has singularities only of ADE type. We obtain an explicit expression for the generating series of the Euler characteristics of the zero-dimensional components in the moduli space of zero-dimensional subschemes on $X$ invariant under the action of $G$. We show that this generating series (up to a suitable rational power of the formal variable) is a holomorphic modular form.
We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a $p$-adic interpolation of the usual Hodge filtration. We relate it with the sheaf of overconvergent Hilbert modular forms.
We prove a generic smoothness result in rigid analytic geometry over a characteristic zero nonarchimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well-adapted to "spreading out" arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.
A torsion free sheaf on a hyperkähler variety $X$ is modular if the discriminant satisfies a certain condition, for example if it is a multiple of $c_2(X)$ the sheaf is modular. The definition is taylor made for torsion-free sheaves on a polarized hyperkähler variety (X,h) which deform to all small deformations of (X,h). For hyperkählers deformation equivalent to $K3^{[2]}$ we prove an existence and uniqueness result for slope-stable modular vector bundles with certain ranks, $c_1$ and $c_2$. As a consequence we get uniqueness up to isomorphism of the tautological quotient rank $4$ vector bund…
We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some étale cover of the moduli stack, up to a finite 2-torsion group.
We develop a full 6-functor formalism for $p$-torsion étale sheaves in rigid-analytic geometry. More concretely, we use the recently developed condensed mathematics by Clausen--Scholze to associate to every small v-stack (e.g. rigid-analytic variety) $X$ with pseudouniformizer $π$ an $\infty$-category $\mathcal D^a_\square(\mathcal O^+_X/π)$ of "derived quasicoherent complete topological $\mathcal O^+_X/π$-modules" on $X$. We then construct the six functors $\otimes$, $\underline{Hom}$, $f^*$, $f_*$, $f_!$ and $f^!$ in this setting and show that they satisfy all the expected compatibilities, s…
We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This le…
A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the u…
Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that …
On the motive of Kummer varieties associated to $Γ_1(7)$ - Supplement to the paper: The modularity of certain non-rigid Calabi-Yau threefolds (by R. Livné and N. Yui)
In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that p-adic Hopf varieties will never have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves.
